Related papers: The sorting index on colored permutations and even…
Let $S_n$ denote the set of permutations of $[n]:=\{1,\cdots, n\}$, and denote a permutation $\sigma\in S_n$ by $\sigma=\sigma_1\sigma_2\cdots \sigma_n$. For $l\ge2$ an integer, let $A^{(n)}_{l;k}\subset S_n$ denote the event that the set…
Connections between $q$-rook polynomials and matrices over finite fields are exploited to derive a new statistic for Garsia and Remmel's $q$-hit polynomial. Both this new statistic $mat$ and another statistic for the $q$-hit polynomial…
Closed-form expressions for the distributions of the order statistics on the spacings between order statistics for the uniform distribution are obtained. This generalizes a result by Fisher concerning tests of significance in the harmonic…
The discrete distribution of the length of longest increasing subsequences in random permutations of $n$ integers is deeply related to random matrix theory. In a seminal work, Baik, Deift and Johansson provided an asymptotics in terms of…
A well known theorem due to Koksma states that for Lebesgue almost every $x>1$ the sequence $(x^n)_{n=1}^{\infty}$ is uniformly distributed modulo one. In this paper we give sufficient conditions for an analogue of this theorem to hold for…
The $r$-rounds Even-Mansour block cipher is a generalization of the well known Even-Mansour block cipher to $r$ iterations. Attacks on this construction were described by Nikoli\'c et al. and Dinur et al., for $r = 2, 3$. These attacks are…
Let $\gamma_n$ be the permutation on $n$ symbols defined by $\gamma_n = (1\ 2\...\ n)$. We are interested in an enumerative problem on colored permutations, that is permutations $\beta$ of $n$ in which the numbers from 1 to $n$ are colored…
It is known that, when $n$ is even, the number of permutations of $\{1,2,\dots,n\}$ all of whose cycles have odd length equals the number of those all of whose cycles have even length. Adin, Heged\H{u}s and Roichman recently found a…
The \emph{coloring number} $\mathrm{col}(G)$ of a graph $G$, which is equal to the \emph{degeneracy} of $G$ plus one, provides a very useful measure for the uniform sparsity of $G$. The coloring number is generalized by three series of…
The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to…
We exhibit a bijection between recently-introduced combinatorial objects known as valid hook configurations and certain weighted set partitions. When restricting our attention to set partitions that are matchings, we obtain three new…
We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough…
We introduce consecutive-pattern-avoiding stack-sorting maps $\text{SC}_\sigma$, which are natural generalizations of West's stack-sorting map $s$ and natural analogues of the classical-pattern-avoiding stack-sorting maps $s_\sigma$…
We extend and generalize many of the enumerative results concerning West's stack-sorting map $s$. First, we prove a useful theorem that allows one to efficiently compute $|s^{-1}(\pi)|$ for any permutation $\pi$, answering a question of…
Pattern avoidance in the symmetric group $S_n$ has provided a number of useful connections between seemingly unrelated problems from stack-sorting to Schubert varieties. Recent work has generalized these results to $S_n\wr C_c$, the objects…
Foata and Zeilberger defined the graphical major index, $\mathrm{maj}'_U$, and the graphical inversion index, $\mathrm{inv}'_U$, for words. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and…
We consider two related problems arising from a question of R. Graham on quasirandom phenomena in permutation patterns. A ``pattern'' in a permutation $\sigma$ is the order type of the restriction of $\sigma : [n] \to [n]$ to a subset $S…
Our main results in this paper are new equidistributions on plane trees and $132$-avoiding permutations, two closely related objects. As for the former, we discover a characteristic for vertices of plane trees that is equally distributed as…
If $G$ is the symmetry group of an uncolored pattern then a coloring of the pattern is semiperfect if the associated color group $H$ is a subgroup of $G$ of index 2. We give results on how to identify and enumerate all inequivalent…
A systematic study of avoidance of mesh patterns of length 2 was conducted by Hilmarsson et al., where 25 out of 65 non-equivalent cases were solved. In this paper, we give 27 distribution results for these patterns including 14…